Chapter Six Discrete Probability Distributions Section 6.3 The Poisson Probability Distribution

A random variable X, the number of successes in a fixed interval, follows a Poisson process provided the following conditions are met 1. The probability of two or more successes in any sufficiently small subinterval is The probability of success is the same for any two intervals of equal length. 3. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping.

EXAMPLE A Poisson Process The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram.

For a sufficiently small interval, the probability of two successes is 0. The probability of insect filth in one region of a candy bar is equal to the probability of insect filth in some other region of the candy bar. The number of successes in any random sample is independent of the number of successes in any other random sample.

EXAMPLEComputing Poisson Probabilities The Food and Drug Administration sets a Food Defect Action Level (FDAL) for various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL level for insect filth in chocolate is 0.6 insect fragments (larvae, eggs, body parts, and so on) per 1 gram. (a)Determine the mean number of insect fragments in a 5 gram sample of chocolate. (b) What is the standard deviation?

Probability Histogram of a Poisson Distribution with  = 1

Probability Histogram of a Poisson Distribution with  = 3

Probability Histogram of a Poisson Distribution with  = 7

Probability Histogram of a Poisson Distribution with  = 15

EXAMPLEPoisson Particles In 1910, Ernest Rutherford and Hans Geiger recorded the number of  -particles emitted from a polonium source in eighth-minute (7.5 second) intervals. The results are reported in the table on the next slide. Does a Poisson probability function accurately describe the number of  -particles emitted? Source: Rutherford, Sir Ernest; Chadwick, James; and Ellis, C.D.. Radiations from Radioactive Substances. London, Cambridge University Press, 1951, p. 172.

The Poisson probability distribution function can be used to approximate binomial probabilities provided the number of trials n > 100 and np < 10. In other words, the number of independent trials of the binomial experiment should be large and the probability of success should be small.

EXAMPLE Using the Poisson Distribution to Approximate Binomial Probabilities According to the U.S. National Center for Health Statistics, 7.6% of male children under the age of 15 years have been diagnosed with Attention Deficit Disorder (ADD). In a random sample of 120 male children under the age of 15 years, what is the probability that at least 4 of the children have ADD?